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[itex](f\star g)(t) = \int_{-\infty}^{\infty}f(\tau)g(t-\tau)d\tau[/itex]

However, I for my own purposes I have invented a similar but different type of "convolution" which has the form

[itex](f\star g)(t) = \int_0^{\infty}f(\tau)g(t/\tau)d\tau[/itex]

So instead of shifting the function g(t) arithmetically, I shift it multiplicatively with [itex]\tau[/itex] before getting the product and integrating, and I only integrate over the positive domain.

What would be an appropriate name or description of this type of function? Is it discussed somewhere?